Resistive exoskeleton control design framework

ABSTRACT

A resistive exoskeleton control system has a controller generating a positive resistance by shaping a closed loop integral admittance of a coupled human exoskeleton system wherein a frequency response magnitude of the integral admittance is lower than that of a natural human joint for desired frequencies of interest and generating an assistance ratio of approximately zero for the desired frequencies of interest.

TECHNICAL FIELD

The present application generally relates to exoskeletons, and, moreparticularly, to a control design framework for designing exoskeletoncontrollers that resist human motion resulting in motion reduction andtorque amplification.

BACKGROUND

Exoskeletons are wearable mechanical devices that may possess akinematic configuration similar to that of the human body and that mayhave the ability to follow the movements of the user's extremities.Powered exoskeletons may be designed to produce contact forces to assistthe user in performing a motor task. In the past, a majority of theresearch on exoskeletons generally has focused on providing assistanceto human limbs, where the assistance may potentially allow humans tocarry loads with less effort (H. Kazerooni and R. Steger, “Berkeleylower extremity exoskeleton,” ASME J. Dyn. Syst., Meas., Control, vol.128, pp. 14-25. 2006.) and (L. M. Mooney, E. I. Rouse, and H. M. Herr,“Autonomous exoskeleton reduces metabolic cost of human walking duringload carriage,” Journal of Neuroengineering and Rehabilitation, vol, 11,no. 80. 2014.); walk faster (S. Lee and Y. Sankai, “Virtual impedanceadjustment in unconstrained motion for an exoskeletal robot assistingthe lower limb,” Advanced Robotics, vol. 19, no. 7, pp. 773-795, 2005.)and (G. S. Sawicki and D. P. Ferris, “Mechanics and energetics of levelwalking with powered ankle exoskeletons,” J. Exp. Biol., vol. 211, no.Pt. 9, pp. 1402-1413, 2008.) and provide torque assist to joints (J. E.Pratt, B. T. Krupp, C. J. Morse, and S. H. Collins, “The RoboKnee: Anexoskeleton for enhancing strength and endurance during walking,” inProc. IEEE Int. Conf. Robotics and Automation (ICRA), 2004, pp.2430-2435.) and (K. E. Gordon, C. R. Kinnaird, and D. P. Ferris,“Locomotor adaptation to a soleus emg-controlled antagonisticexoskeleton,” J. Neurophysiol., vol. 109, no. 7, pp. 1804-1814, 2013.).

Exoskeletons may be used to provide resistance to human motion. Byproviding resistance to human motion, the exoskeletons may be used forexercise and rehabilitation applications. Resistance training with upperbody exoskeletons has been used in the past. (Z. Song and Z. Wang,“Study on resistance training for upper-limb rehabilitation using anexoskeleton device,” in Proc. IEEE Int'l Conf. Mechatronics andAutomation, 2013, pp. 932-938.); (Z. Song, S. Guo, M. Pang, S. Zhang, N.Xiao, B. Gao, and L. Shi, “Implementation of resistance training usingan upper-limb exoskeleton rehabilitation device for elbow joint,” 3.Med. Bio. Engg., vol. 34, no. 2, pp. 188-196, 2014.) and (T.-M. Wu andD.-Z. Chen, “Biomechanical study of upper-limb exoskeleton forresistance training with three-dimensional motion analysis system,” J.Rehabil. Res. Dev., vol. 51, no. 1, pp. 111-126, 2014.). Upper bodyexoskeletons that may resist human motion with applications to tremorsuppression have been used for rehabilitation (E. Rocon and J. L. Pons,Exoskeletons in Rehabilitation Robotics:Tremor Suppression. SpringerTracts in Advanced Robotics, 2011, pp. 67-98.). In 2013, NASA introducedthe X1 exoskeleton ((2013) Nasa's x1 exoskeleton.http://www.nasa.gov/offices/oct/home/feature_exoskeleton.html). The X1exoskeleton may be capable of providing both assistance and resistanceto the joints in the leg. The X1 exoskeleton may be used as an exercisedevice that may improve the health of astronauts during their time inspace, and may also be used for rehabilitation applications.

Even with previous efforts in exoskeleton design and implementation,there continues to be a need for a resistive exoskeleton control designframework that provides exoskeleton control parameters that achievedesired resistance. Therefore, it would be desirable to provide a systemand method that overcome the above. The system and method would providea resistive exoskeleton control design framework that providesexoskeleton control parameters that achieve desired resistance whileensuring that the resulting coupled system dynamics are both stable andpassive.

SUMMARY

In accordance with one embodiment, a resistive exoskeleton controlsystem is disclosed. The control system has a controller shaping aclosed loop integral admittance of a coupled human exoskeleton systemwherein a frequency response magnitude of the integral admittance islower than that of a natural human joint for desired frequencies ofinterest and generating an assistance ratio of approximately zero overthe desired frequencies of interest.

In accordance with one embodiment, a resistive exoskeleton controlsystem is disclosed. The resistive exoskeleton control system has acontroller shaping a closed loop integral admittance of a coupled humanexoskeleton system wherein a frequency response magnitude of theintegral admittance is lower than that of a natural human joint andgenerating an assistance ratio of approximately zero for desiredfrequencies of interest, wherein the controller being stable andpassive.

In accordance with one embodiment, an exoskeleton control system isdisclosed. The exoskeleton control system has a controller generating apositive resistance and approximately zero assistance by shaping aclosed loop integral admittance of a coupled human exoskeleton systemover a desired frequency range, wherein the controller being stable andpassive.

BRIEF DESCRIPTION OF THE DRAWINGS

In the descriptions that follow, like parts are marked throughout thespecification and drawings with the same numerals, respectively. Thedrawing figures are not necessarily drawn to scale and certain figuresmay be shown in exaggerated or generalized form in the interest ofclarity and conciseness. The disclosure itself, however, as well as apreferred mode of use, further objectives and advantages thereof, willbe best understood by reference to the following detailed description ofillustrative embodiments when read in conjunction with the accompanyingdrawings, wherein:

FIG. 1 is a perspective view of an exoskeleton device implementing anexemplary admittance shaping controller in accordance with one aspect ofthe present application;

FIG. 2A and FIG. 2B show representations of an exemplary onedegree-of-freedom (1-DOF) coupled human-exoskeleton system with rigidcoupling (FIG. 2A) and soft coupling (FIG. 2B) in accordance with oneaspect of the present application;

FIG. 3 is a block diagram summarizing equations for the 1-DOF coupledhuman-exoskeleton system with rigid coupling and soft coupling inaccordance with one aspect of the present application;

FIG. 4A and FIG. 4B are block diagrams of illustrative coupledhuman-exoskeleton systems with the exoskeleton controller U_(e)(s) inaccordance with one aspect of the present application;

FIG. 5A is an illustrative graph of Assistance and Resistance of a 1-DOFassistive exoskeleton represented in terms of Integral Admittance X(s)in accordance with one aspect of the present application;

FIG. 5B is an illustrative graph of Assistance and Resistance of a 1-DOFassistive exoskeleton represented in terms of an assistance functionAT(w) and a resistance function RF(ω) in accordance with one aspect ofthe present application;

FIGS. 6A-6C are illustrative graphs showing derived control parametersfor different desired resistance ratios R_(d) with the coupledstability, passivity, and zero assistance constraints in accordance withone aspect of the present application;

FIG. 7 shows an illustrative series of Nyquist plots of L_(heu)(s) forthe control parameters in FIGS. 6(a)-6(c) that achieve both coupledstability and passivity in accordance with one aspect of the presentapplication;

FIG. 8A and FIG. 8B are illustrative integral admittance magnitude andphase plots for derived coupled human-exoskeleton systems in accordancewith one aspect of the present application; and

FIG. 9 is an illustrative functional block diagram of an exemplaryframework used for resistive exoskeleton control in accordance with oneaspect of the present application.

DESCRIPTION OF THE INVENTION

The description set forth below in connection with the appended drawingsis intended as a description of presently preferred embodiments of thedisclosure and is not intended to represent the only forms in which thepresent disclosure may be constructed and/or utilized. The descriptionsets forth the functions and the sequence of steps for constructing andoperating the disclosure in connection with the illustrated embodiments.It is to be understood, however, that the same or equivalent functionsand sequences may be accomplished by different embodiments that are alsointended to be encompassed within the spirit and scope of thisdisclosure.

Embodiments of the disclosure provide a control design framework fordesigning resistive exoskeleton controllers that may resist human jointmotion. Resistance in regards to exoskeleton controllers may be definedas the decreasing of the frequency response magnitude profile of theintegral admittance of the coupled human-exoskeleton system below thatof the normal human limb. An exoskeleton controller may be resistive ifthe controller increases the impedance and decreases the admittance ofthe coupled human-exoskeleton joint. A resistive exoskeleton controllermay result in motion reduction, i.e., the joint motion amplitude may belower for the same joint torque profile, and torque amplification, i.e.,the joint torque amplitude required to achieve the same joint motion maybe larger.

The present control design framework may modify the coupled system jointdynamics such that system admittance may be decreased. More precisely,the coupled joint dynamics may be characterized by the frequencyresponse magnitude profile of the coupled system integral admittance(torque-to-angle relationship), and resistance may be achieved when thefrequency response magnitude profile of the integral admittance of thecoupled system may be lower than that of the natural human joint for allfrequencies of interest. The resistive control design framework mayprovide exoskeleton control parameters that may ensure that the coupledsystem is stable and passive while achieving the desired resistance. Thepresent control design framework may be formulated as a constrainedoptimization problem, with the objective of finding exoskeleton controlparameters that achieve a desired resistance while satisfying coupledstability and passivity constraints.

The present control design framework may provide resistive exoskeletonsthat may be used in rehabilitation applications for resistance training,and may be used by non-pathological humans for physical exercises andmuscle building. Embodiments of the control design framework may allow asingle exoskeleton device to emulate different physical trainingconditions with increased weight, increased damping (walking in sand orwater), increased stiffness (walking uphill), and any combinationsthereof. Therefore, instead of moving to different conditions orlocations for physical training, a human subject may use a single deviceto emulate the different conditions in a single location of theirchoosing.

Embodiments of the control design framework may be modified to designexoskeleton controllers that provide assistance and avoid resistance.The controllers that assist at some frequencies and resist at some otherfrequencies may also be designed using the disclosed framework. Theshape of the response curve of the integral admittance of the disclosedcoupled system may be shaped to achieve a variety of different desireddynamic responses for the human limb.

It should be noted that while a framework for a one degree-of-freedom(1-DOF) exoskeleton is disclosed herein, embodiments of the novelframework may be extended to multiple degrees-of-freedom (DOF)exoskeletons. The disclosed framework is not limited to lower-limbexoskeletons and may be extended to upper-limb exoskeletons, as well aswhole body exoskeleton devices with resistive controllers at each jointthat may help in physical training for the whole body. The disclosedframework may be extended to task-level resistance instead ofjoint-level resistance. For example, the exoskeleton controllers may bedesigned to resist the motion of the foot (task-level output) ratherthan resist the hip, knee and ankle (joint-level outputs) joint motions.

The system parameters of the coupled human-exoskeleton system used inthe analysis and experimental results presented in the exemplaryembodiments of the disclosure may be seen in Table 1 shown below. Thehuman limb data corresponds to the leg of a human whose weight may beapproximately 65 kg and height approximately 1.65 m. In the exemplaryembodiments of the disclosure, the knee may be assumed to be locked andall parameters may be computed for the hip joint. The moment of inertiaI_(h) may be obtained from Cadaver data provided in “Biomechanics andMotor Control of Human Movement” by D. A. Winter (4^(th) Edition, Wiley,2009, p. 86), and may be scaled to the human weight and height. Thejoint damping coefficient may be taken from “Passive visco-elasticproperties of the structures spanning the human elbow joint,” by K. C.Hayes and H. Hatze (European Journal Applied Physiology, vol. 37, pp.265-274, 1977), and the joint stiffness coefficient may be obtainedusing k_(h)=I_(h)ω² _(nh) where the natural frequency ω_(nh) may beobtained from “Mechanics and energetics of swinging the human leg” by J.Doke, J. M. Donelan, and A. D. Kuo (Journal of Experimental Biology,vol. 208, pp. 439-445, 2005).

TABLE 1 COUPLED HUMAN-EXOSKELETON SYSTEM PARAMETERS Parameters SymbolValue Human Leg Mass m_(h) 10.465 kg (locked knee) Human Leg Lengthl_(h) 0.875 m Human Leg I_(h) 3.381 kg · m² Moment of Inertia Human HipJoint b_(h) 3.5 N · m · s/rad Damping Coefficient Human Hip Joint k_(h)54.677 N · m/rad Stiffness Coefficient Human Leg Natural ω_(nh) 4.021rad/s Angular Frequency Exoskeleton Arm I_(c) 0.01178 kg · m² Moment ofInertia Exoskeleton Joint b_(ε) 0.34512 N · m · s/rad DampingCoefficient Exoskeleton Joint k_(c) 0.33895 N · m/rad StiffnessCoefficient Coupling b_(c) 9.474 N · m · s/rad Damping CoefficientCoupling k_(c) 1905.043 N · m/rad Stiffness Coefficient

The exoskeleton parameters listed in Table 1 may be obtained from systemidentification experiments on a 1-DOF hip exoskeleton shown in FIG. 1,which is described in greater detail below. The coupling parameterslisted in Table 1 may be obtained with the assumption that the couplingparameters second-order dynamics with the exoskeleton inertia I_(e) mayhave a damping ratio ζ_(c)=1 and natural frequency ω_(nC)=100 ω_(nh),where ω_(nh) is the natural frequency of the human limb.

Referring now to the figures, FIG. 1 shows an embodiment of a StrideManagement Assist (SMA) exoskeleton device 10 in accordance withembodiments of the invention. The SMA device 10 may have two 1-DOF hipjoints 12. In the embodiment shown, a torso support 14 may be added tothe SMA device 10 to provide greater damping and to reduce oscillations.The coupling between an exoskeleton and a human limb 16 may play acritical role in determining the performance of the exoskeleton 10, andthe coupling may be either rigid 18 or soft 20 as shown in FIGS. 2A-2Band in FIG. 3. In the case of rigid coupling 18 as shown in FIG. 2A,there may be no relative motion between the human limb 16 and theexoskeleton 10, whereas in the case of soft coupling 20 in FIG. 2B, thehuman limb 16 and the exoskeleton 10 may move relative to each other. Inactual implementations of an exoskeleton attached to a limb there may bea soft coupling due to muscle, skin tissue, fat layers, and other bodysubstances between the bone and the exoskeleton device. The soft couplemay be modeled in embodiments with a linear torsional spring with acoefficient k, and a linear torsional damper with coefficient b_(e) asshown in FIG. 3.

As shown in FIG. 3, a coupled human-exoskeleton system with softcoupling may be modeled with second-order linear models to represent thejoint dynamics of a human as shown in Equation 1 below, and theexoskeleton in Equation 2. The moment of inertia, joint damping andjoint stiffness of the 1-DOF human joint may be given by I_(h), b_(h),k_(h) respectively, and that for the exoskeleton may be given by {I_(e),b_(e), k_(e)}. The coupling damping and stiffness coefficients may begiven by b_(e), k_(e), respectively.

The linear equations of motion of an isolated 1-DOF human joint of anexemplary embodiment of the disclosure may be given by

I _(h){umlaut over (θ)}_(h)(i)+b _(h){dot over (θ)}_(h)(t)+k_(h)θ_(h)(t)=τ_(h)(t)₁   (1)

Where θ_(h)(t) is the joint angle trajectory, I_(h), b_(h), k_(h) is theassociated moment of inertia, joint damping coefficient and jointstiffness coefficient respectively, and τ_(h)(t) is the joint torquetrajectory. The stiffness term k_(h)θ_(h)(t) may include the linearizedgravitational terms. Similarly, the linear equations of motion of anisolated 1-DOF exoskeleton may be given by:

I _(e){umlaut over (θ)}_(e)(t)+b _(e){dot over (θ)}_(e)(t)+k_(e)θ_(e)(t)=τ_(e)(t),   (2)

where θ_(h)(t) is the joint angle trajectory, I_(e), b_(e), k_(e) is theassociated moment of inertia, joint damping coefficient and jointstiffness coefficient respectively, and τ_(h)(t) is the joint torquetrajectory.

The linear equations of motion of a coupled human exoskeleton systemwith soft coupling may be given by:

I _(h){umlaut over (θ)}_(h)(t)+b _(h){dot over (θ)}_(h_l () t)+k_(h)θ_(h)(t)=τ_(h)(t)−τ_(e)(t),   (3)

I _(e){umlaut over (θ)}_(e)(t)+b _(e){dot over (θ)}_(e)(t)+k_(e)θ_(e)(t)=τ_(e)(t)+τ_(e)(t),   (4)

where τ_(e) is the coupling joint torque given by:

τ_(e)(t)=b _(e)({dot over (θ)}_(h)(t)−{dot over (θ)}_(e)(t))+k_(e)(θ_(h)(t)−θ_(e)(t)),   (5)

For the linear human joint dynamics in Equation (1), the impedance (N.Hogan and S. O. Buerger, Impedance and Interaction Control, Robotics andAutomation Handbook. CRC Press, LLC., 2005, ch. 19) transfer functionZ_(h)(s) may be given by

$\begin{matrix}{{{Z_{h}(s)} = {\frac{\tau_{h}(s)}{\Omega_{h}(s)} = \frac{{I_{h}s^{2}} + {b_{h}s} + k_{h}}{s}}},} & (6)\end{matrix}$

and the admittance (N. Hogan and S. O. Fuerger, Impedance andInteraction Control, Robotics and Automation Handbook. CRC Press, LLC.,2005, ch. 19) transfer function Y_(h)(s) may be given by:

$\begin{matrix}{{{Y_{h}(s)} = {\frac{\Omega_{h}(s)}{\tau_{h}(s)} = \frac{s}{{I_{h}s^{2}} + {b_{h}s} + k_{h}}}},} & (7)\end{matrix}$

where Q_(h)(s) is the Laplace transform of {umlaut over (θ)}_(h)(t), andτ_(h)(s) is the Laplance transform of τ_(h)(t). For a linear system, itsimpedance may be the inverse of its admittance and vice-versa, as it canbe seen in Equations 6-7.

The integral admittance transfer function X_(h)(s) may be defined as theintegral of the admittance transfer function and may be given by:

$\begin{matrix}{{{X_{h}(s)} = {\frac{\Theta_{h}(s)}{\tau_{h}(s)} = \frac{1}{{I_{h}s^{2}} + {b_{h}s} + k_{h}}}},} & (8)\end{matrix}$

where Θ_(h)(s) is the Laplace transform of θ_(h)(t). The admittanceY_(h)(s) maps torque to angular velocity, while the integral admittanceX_(h)(s) maps torque to angle. The integral admittance may be usedextensively in the further sections of this disclosure.

In embodiments described in this disclosure the human joint,exoskeleton, and coupling element may be treated as three isolatedsystems, and their corresponding impedance and admittance transferfunctions may be written as follows. The admittance transfer function ofan isolated human joint Y_(h)(s) may be given by Equation 7, while theadmittance transfer function of an isolated exoskeleton Y_(e)(s) may begiven by:

$\begin{matrix}{\mspace{20mu} {{{Y_{e}(s)} = {\frac{\Omega_{\text{?}}(s)}{\tau_{\text{?}}(s)} = \frac{1}{{I_{\text{?}}s^{2}} + {b_{\text{?}}s} + k_{\text{?}}}}},{\text{?}\text{indicates text missing or illegible when filed}}}} & (9)\end{matrix}$

and the impedance transfer function of an isolated coupling elementZ_(e)(s) may be given by:

$\begin{matrix}{\mspace{20mu} {{{Z_{c}(s)} = {\frac{\tau_{\text{?}}(s)}{\Omega_{\text{?}}(s)} = \frac{{b_{\text{?}}s} + k_{\text{?}}}{s}}},{\text{?}\text{indicates text missing or illegible when filed}}}} & (10)\end{matrix}$

where Ω_(e)(s)=Ω_(h)(s)−Ω_(e)(s) is the Laplace transform of the angularvelocity of the coupling element.

As disclosed herein exoskeleton controllers may be designed to modifythe coupled system joint dynamics, i.e., the joint impedance,admittance, and integral admittance of the coupled human exoskeletonsystem. The following is a derivation of an embodiment of theclosed-loop dynamics of a coupled human-exoskeleton system with anexoskeleton controller, and presents the coupled stability and passivityconditions.

For any exoskeleton control transfer function U_(e)(s) that feeds backexoskeleton joint angular velocity transfer function Ω_(e)(s), theclosed-loop coupled human-exoskeleton system may be represented as ablock diagram in FIG. 4A. The analysis presented below may be applicableto any general exoskeleton controller U_(e)(s), and the specificexoskeleton control structure used in this disclosure may be seen below.

The outlined region 22 in FIG. 4A containing Z_(e)(s), Y_(e)(s) andU_(e)(s) may be reduced to a single transfer function Z_(ceu)(s) whichmay be given by:

$\begin{matrix}{{{Z_{ceu}(s)} = \frac{Z_{c}(s)}{1 + {{Z_{c}(s)}{Y_{cu}(s)}}}},} & (11)\end{matrix}$

as shown in FIG. 4B where Y_(eu)(s) may be given by:

$\begin{matrix}{{Y_{cu}(s)} = {\frac{Y_{e}(s)}{1 - {{Y_{e}(s)}{U_{e}(s)}}}.}} & (12)\end{matrix}$

The loop transfer function L_(heu)(s) that may be needed to evaluate thestability of the feedback system shown in FIG. 4B may be given by:

L _(heu)(s)=Y _(h)(s)Z _(eus)(s)   (13)

and the feedback system gain margin GM may be given by:

$\begin{matrix}{{{{GM}\left( L_{heu} \right)} = \frac{1}{{L_{heu}\left( {j\omega}_{e} \right)}}},} & (14)\end{matrix}$

where ω_(e) is the phase-crossover frequency when the phase ofL_(heu)(s) is 180°, i.e.,) /L_(heu)(jω_(e))=180°. The gain marginGM(L_(heu)) may give the maximum positive gain exceeding when theclosed-loop system becomes unstable. Therefore, in order for the coupledhuman-exoskeleton system shown in FIG. 4B to be stable, the followingcondition may generally needs to be satisfied:

GM(L _(heu))>1.   (15)

From FIG. 4B, the overall closed-loop admittance Y_(heu)(s) of thecoupled human-exoskeleton system with the exoskeleton controllerU_(e)(s) may be given by:

$\begin{matrix}{\mspace{20mu} {{Y_{heu}(s)} = {{\frac{Y_{h}(s)}{1 + {{Y_{h}(s)}\text{?}(s)}}.\text{?}}\text{indicates text missing or illegible when filed}}}} & (16)\end{matrix}$

and the corresponding closed-loop integral admittance X_(heu)(s) of thecoupled human-exoskeleton system may be given by:

$\begin{matrix}{\mspace{20mu} {{{X_{heu}(s)} = {\frac{Y_{heu}(s)}{s} = \frac{X_{h}(s)}{1 + {{Y_{h}(s)}\text{?}(s)}}}},{\text{?}\text{indicates text missing or illegible when filed}}}} & (17)\end{matrix}$

where X_(h)(s)=Y_(h)(s)/s as shown in Equation 8. It should be notedthat the natural human joint dynamics of a second-order may be shown inEquation 1, while the coupled human-exoskeleton joint dynamics shown inEquation 3-5 is of a fourth-order. However, with high coupling stiffnessand damping, the coupled system dynamics may be predominantly of asecond-order. The order of the closed-loop coupled system may depend onthe order of the exoskeleton controller U_(e)(s).

In addition to coupled stability, an important requirement fordynamically interacting systems may be coupled passivity (J. E. Colgate,“The control of dynamically interacting systems,” PhD. dissertation,Massachusetts Institute of Technology, Cambridge, Mass., 1988). Coupledpassivity may ensure that the coupled human-exoskeleton system does notbecome unstable when in contact with any passive environment (J. E.Colgate and N. Hogan, “An analysis of contact instability in terms ofpassive physical equivalents,” in Proc. IEEE Int. Conf. Robotics andAutomation (ICRA), 1989, pp. 404-409). A linear time-invariant systemmay be said to be passive (J. E. Colgate, “The control of dynamicallyinteracting systems,” Ph.D. dissertation, Massachusetts Institute ofTechnology, Cambridge, Mass., 1988) when the impedance transfer functionZ(s) satisfies the following conditions:

-   -   1) Z(s) has no poles in the right-hand half of the complex        plane; and    -   2) Z(s) has a Nyquist plot that lies wholly in the right-hand        half of the complex.        The first condition generally requires Z(s) to be stable, while        the second condition generally requires the phase of Z(s) to lie        within −90° and 90° for all frequencies (J. E. Colgate, “The        control of dynamically interacting systems,” Ph.D. dissertation,        Massachusetts Institute of Technology, Cambridge, Mass., 1988),        i.e., /Z(jω)∈[−90°, 90°]. This, in turn, may enforce that the        phase of the system admittance /Y(jω)∈[−90°, 90°] and the phase        of the system integral admittance /X(jω)∈[−180 °, 0°].

Therefore, in order for a stable coupled human-exoskeleton systemsatisfying Equation 15 to be passive, the following condition may needto be satisfied:

/ X _(heu)(jω)∈[−180 °, 0°] ∀ω.   (18)

A novel control design framework may be disclosed below that may shapethe frequency response magnitude of the closed-loop integral admittanceX_(heu)(s) of the coupled human exoskeleton system in Equation 17 suchthat the 1-DOF human joint motion may be resisted. In this framework,the magnitude may be chosen for shaping the integral admittance profile,while the phase may be used to evaluate the passivity of the coupledsystem.

In order to design the shape of |X_(heu)(jω)|, an objective for theexoskeleton may need to be defined. In an exemplary embodiment, anobjective may be to provide resistance and avoid assistance. In order todefine the shape of |X_(heu)(jω)| that provides resistance and avoidsassistance, the resistance and assistance may need to be defined in aclear and quantitative way. Below, presents conceptual and quantitativedefinitions of resistance and assistance using the frequency responsemagnitude of the integral admittance, followed by a description of thedesired characteristics of a resistive exoskeleton, and a constrainedoptimization formulation that shapes the closed-loop integral admittancesuch that the desired resistance is achieved, while guaranteeing coupledstability and passivity.

The following definitions for resistance and assistance may be use inaccordance with embodiments of the disclosure. Definition 1: In anexemplary embodiment of the present disclosure a 1-DOF human joint maybe said to be “resisted” by an exoskeleton if the frequency responsemagnitude of the integral admittance of the coupled human-exoskeletonsystem is lesser than that of the natural human for all frequencies ofinterest, i.e., |X_(heu)(jω)|<|X_(h)(jω)|, ∀ω∈[0ω_(f)] where ω_(f) isthe upper bound for the frequencies of interest. When a joint isresisted as per Definition 1, the same joint torques may produce a jointmotion whose amplitude is smaller than that of the natural joint, and istermed as motion reduction. On the other hand, the same joint motion maybe achieved in the resisted joint with a torque profile whose amplitudeis larger than that required for the natural joint. This may be termedas torque amplification.

Definition 2: A 1-DOF human joint may be said to be assisted by anexoskeleton if the frequency response magnitude of the integraladmittance of the coupled human-exoskeleton system is greater than thatof the natural human for all frequencies of interest, i.e.,|X_(heu)(jω)|>|X_(h)(jω)|, ∀ω∈[0, ω_(f)]. Similar to motion reductionand torque amplification that result from resistance, assistance mayproduce their opposite effects, i.e., motion amplification and torquereduction.

As may be seen in FIGS. 5A and 5B, plots of assistance and resistance ofa 1-DOF assistive exoskeleton represented in terms of: (a) integraladmittance X(s), and (b) assistance function AF(ω) and Resistance RF(ω).The magnitude of the frequency response of the integral admittance of anatural human joint may be denoted by |X_(h)(jω)|, and that of theclosed-loop integral admittance of the coupled human-exoskeleton systemmay be denoted by |X_(heu)(jω)|.

As shown in FIG. 5A, a frequency response magnitude plot of a naturalhuman and a hypothetical coupled human exoskeleton system whoseparameters are listed in Table 1 may be seen. As per definitions 1 and 2above, region I may represent the frequencies where there is resistance,i.e., |X_(heu)(jω)|<|X_(h)(jω)|, and region II may represent thefrequencies where there is assistance, i.e., |X_(heu)(jω)|>|X_(h)(jω)|.

FIG. 5B shows that at each frequency ω, the exoskeleton behavior may beeither resistive or assistive. Based on this observation, the resistancefunction RF(ω) may be defined as

$\begin{matrix}{{{\mathcal{F}}(\omega)} = \left\{ \begin{matrix}0 & {{{if}\mspace{14mu} {{X_{heu}({j\omega})}}} \geq {{X_{h}({j\omega})}}} \\\frac{{{X_{h}({j\omega})}} - {{X_{heu}({j\omega})}}}{{X_{h}({j\omega})}} & {{{if}\mspace{14mu} {{X_{heu}({j\omega})}}} < {{X_{h}({j\omega})}}}\end{matrix} \right.} & (19)\end{matrix}$

and the assistance function AF(ω) may be defined as:

$\begin{matrix}{{{\mathcal{F}}(\omega)} = \left\{ \begin{matrix}\frac{{{X_{heu}({j\omega})}} - {{X_{h}({j\omega})}}}{{X_{h}({j\omega})}} & {{{if}\mspace{14mu} {{X_{heu}({j\omega})}}} \geq {{X_{h}({j\omega})}}} \\0 & {{{if}\mspace{14mu} {{X_{heu}({j\omega})}}} < {{X_{h}({j\omega})}}}\end{matrix} \right.} & (20)\end{matrix}$

At any frequency ω, the resistance function RF(ω)∈[0,1], and theassistance function AF(ω)∈[0, ∞]. When the coupled human-exoskeletonjoint dynamics may be identical to the natural human joint dynamics,i.e., |X_(heu)(jω)|=|X_(h)(jω)|, then RF(ω)=AF(ω)=0, ∀ω. The upper boundRF(ω)=1 is achieved when |X_(heu)(jω)|=0, and the upper bound AF(ω)=∞may be achieved when |X_(heu)(jω)|−1. Although both these cases aremathematically valid, these cases are generally not realistic.

In specific embodiments of the disclosure, it may be important to notethat the exoskeleton may either only resist or only assist at anyparticular frequency ω for a single joint, which may be seen from FIG.5A. Using the resistance function RF(ω) and the assistance functionAF(ω), the following quantitative metrics for resistance and assistanceof a 1-DOF assistive exoskeleton, namely resistance ratio and assistanceratio may be defined.

Definition 3: Resistance Ratio R may be defined as the average value ofthe resistance function RF(ω) over a range of frequencies [0,ω_(f)] andmay be given by:

$\begin{matrix}{ = {\frac{1}{\omega_{f}}{\int_{0}^{w_{f}}{{{\mathcal{F}}(\omega)}\ {{\omega}.}}}}} & (21)\end{matrix}$

Definition 4: Assistance Ratio A may be defined as the average value ofthe assistance function AF(ω) over a range of frequencies [0,ω_(f)], andmay be given by:

$\begin{matrix}{ = {\frac{1}{\omega_{f}}{\int_{0}^{w_{f}}{{{\mathcal{F}}(\omega)}\ {{\omega}.}}}}} & (22)\end{matrix}$

Similar to the resistance and assistance functions, the resistance ratioR∈[0, 1] and the assistance ratio A∈[0, ∞]. As described above, theupper bounds R=1 and A=∞ may be achieved only if |X_(heu)(jω)|=∞ and|Xa(jω)|=0 respectively ∀ω. Although these bounds may be mathematicallyvalid, the bounds may not be realistic for any proper integraladmittance transfer function. With the above definitions of resistanceand assistance, the below section of may enumerate embodiments ofdesired characteristics of a resistive exoskeleton.

An objective of embodiments of the resistive exoskeleton may be toprovide resistance to any human motion while not be assisting anymotion. However, it may be vital to ensure that the coupledhuman-exoskeleton system is also stable. Furthermore, coupled passivityas defined in “The control of dynamically interacting systems,” by J. E.Colgate (Ph.D. dissertation, Massachusetts Institute of Technology,Cambridge, Mass., 1988) may also be essential since coupled passivitymay guarantee stability even when the coupled human-exoskeleton systeminteracts with any passive environment.

Therefore, the necessary desired characteristics of a 1-DOF resistiveexoskeleton may be listed as follows:

1) Coupled Stability, i.e., GM(L _(heu))>1   (Eq. 15);

2) Coupled Passivity, i.e., / X _(heu)(jω)∈[−180 °,0°], ∀ω  (Eq. 18);

3) Positive Resistance, i.e. R>0   (Eq. 21); and

4) No Assistance, i.e. A=0   (Eq. 22).

The above characteristics may be the necessary desired characteristicsof a 1-DOF resistive exoskeleton. However, more characteristics may beadded to the list depending on the task and the desired goals of theexoskeleton implementation.

The preceding sections may have provided the metrics to evaluateresistance and enumerated the desired characteristics of a resistiveexoskeleton. Now, embodiments of designs for an exoskeleton controllerU_(e)(s) that shapes the closed-loop integral admittance of the coupledhuman-exoskeleton system based on these metrics may be disclosed below.

Any exoskeleton control law for τ_(e)(t) may produce an exoskeletondynamics given by Equation 4, and hence given a desired exoskeletondynamics, one can derive a corresponding controller. If the desiredexoskeleton dynamics may be given by a desired moment of inertia I_(c)^(d), a desired joint damping coefficient b_(e) ^(d) and a desired jointstiffness coefficient k_(e) ^(d), then the exoskeleton torque τ_(e)required to achieve the desired exoskeleton dynamics may be given by

τ_(e)(t)=I _(e) −I _(e) ^(d)){umlaut over (θ)}_(e)(t)+(b _(e) −b _(e)^(d)){dot over (θ)}_(e)(t)+(k _(e) −k _(e) ^(d)θ_(e)(t),   (23)

It can be easily verified that the control law in Equation 23 may reducethe exoskeleton dynamics in Equation 2 to:

I _(e) ^(d){umlaut over (θ)}_(e)(t)+b _(e) ^(d){dot over (θ)}_(e)(t)+k_(e) ^(d)θ_(e)(t)   (24)

as desired. The exoskeleton controller U_(e)(s) corresponding to thecontrol law in Equation 23 that feeds back angular velocity Ω_(e)(s) maybe given by:

$\begin{matrix}{{{U_{e}(s)} = \frac{{K_{\alpha}s^{2}} + {K_{\omega}s} + K_{\theta}}{s}},} & (25)\end{matrix}$

where K_(o)=I_(e)−I_(e) ^(d), K_(ω)=b_(e)−b_(e) ^(d), and K₀=k_(e)−k_(e)^(d) are the feedback gains on angular acceleration {umlaut over(θ)}_(e), angular velocity {dot over (θ)}_(e) and angle θ_(e)respectively.

The control transfer function U_(e)(s) shown in Equation 25 may becharacterized by three control parameters, namely, K_(θ), K_(ω), andK_(α). These parameters may affect the closed-loop integral admittanceX_(heu)(s), and they may be chosen such that the frequency responsemagnitude of the closed-loop integral admittance X_(heu)(s) may beshaped such that the desired resistance R_(d) is achieved.

Given a desired resistance ratio R_(d), an optimal set of controlparameters of the 1-DOF coupled human-exoskeleton system in Equation 17may be obtained using the following constrained optimization problem:

$\begin{matrix}{{\underset{\{{K_{\theta},K_{\omega},K_{\alpha},\omega_{lo}}\}}{minimize}{{ - _{d}}}^{2}}{{{{subject}\mspace{14mu} {to}\mspace{14mu} {{GM}\left( L_{heu} \right)}} > 1},\mspace{110mu} {\frac{/{X_{heu}({j\omega})}}{ = 0.} \in {\left\lbrack {{{- 180}{^\circ}},{0{^\circ}}} \right\rbrack \mspace{14mu} {\forall\omega}}}}} & (26)\end{matrix}$

FIGS. 6A-6C illustrate the derived optimal control parameters for thedifferent desired resistance ratios R_(d) with the coupled stability,passivity and zero assistance constraints. The passive exoskeletonwithout any resistive control already resulted in a resistance ratio ofR=0.0177, and hence the optimization results in FIG. 6 generallycorrespond to desired resistance ratio R_(d)≧0.02.

FIG. 7 shows a set of Nyquist plots of L_(heu)(s) corresponding to someof the optimal control parameters shown in FIG. 6 that achieve bothcoupled stability and passivity. None of the Nyquist plots encircle−1+j0, indicating stable coupled human-exoskeleton systems. The integraladmittance magnitude and phase plots of some of the derived coupledhuman-exoskeleton systems may be shown in FIGS. 8(a) and 8(b). It may beseen from FIG. 8(b) that the coupled systems, whose phase plots areshown satisfy the passivity condition in Equation 18, i.e.,/X_(heu)(jω)∈[−180 °, 0°], ∀ω. Thus, these embodiments of the coupledhuman-exoskeleton system with optimal control parameters that achievecoupled stability and passivity for FIG. 8(a) with integral admittancemagnitude |X_(heu)(jω)|, and for FIG. 8(b) with integral admittancephase /X_(heu) .

FIG. 9 is a functional block of an embodiment of a control frameworkthat executes the resistive exoskeleton controller using the optimalparameters from Equation 26. As shown in FIG. 9, a Kalman filter may beused to estimate the exoskeleton joint angle {umlaut over (θ)}_(e),angular velocity {dot over (θ)}_(e) and angular acceleration {umlautover ({circumflex over (θ)})}_(e) that may be needed to implement thecontrol law in Equation 25. Embodiments of the online Kalman filterimplementation may be based on filter models found in “Kalman filterestimation of angular velocity and acceleration: On-lineimplementation,” by P. Canet (McGill University, Montreal, Canada, Tech.Rep. TR-CIM-94-15, November 1994).

While embodiments of the disclosure have been described in terms ofvarious specific embodiments, those skilled in the art will recognizethat the embodiments of the disclosure may be practiced withmodifications within the spirit and scope of the claims.

What is claimed is:
 1. A resistive exoskeleton control systemcomprising: a controller generating a positive resistance by shaping aclosed loop integral admittance of a coupled human exoskeleton system,wherein a frequency response magnitude of the closed loop integraladmittance is lower than that of a natural human joint over desiredfrequencies of interest and generating an assistance ratio ofapproximately zero over the desired frequencies of interest.
 2. Theresistive exoskeleton control system of claim 1, wherein the controllergenerating a loop transfer function has a gain margin greater than
 1. 3.The resistive exoskeleton control system of claim 1, wherein thecontroller has coupled passivity.
 4. The resistive exoskeleton controlsystem of claim 3, wherein a phase of the closed loop integraladmittance is defined by /X_(heu)(jω)∈[−180 °, 9°[∀ω.
 5. The resistiveexoskeleton control system of claim 1, wherein the controller generatinga control transfer function defined by${{U_{e}(s)} = \frac{{K_{\alpha}s^{2}} + {K_{\omega}s} + K_{\theta}}{s}},$wherein K_(o)=I_(e)−I_(e) ^(d, K) _(ω)=b_(e)−b_(e) ^(d), andK₀=k_(e)−k_(e) ^(d) are the feedback gains on angular acceleration{umlaut over (θ)}_(e), angular velocity {dot over (θ)}_(e) and angleθ_(e) of an exoskeleton device.
 6. The resistive exoskeleton controlsystem of claim 1, wherein the controller generating a positiveresistance ratio over the desired frequencies of interest, wherein thepositive resistance ratio defined by $\begin{matrix}{\mspace{79mu} {{ = {\frac{1}{\omega_{f}}{\int_{0}^{w_{f}}{{{\mathcal{F}}(\omega)}\ {{\omega}.\mspace{14mu} {wherein}}}}}}{{{\mathcal{F}}(\omega)} = \left\{ {\begin{matrix}0 & {{{if}\mspace{14mu} {{X_{heu}({j\omega})}}} \geq {{X_{h}({j\omega})}}} \\\frac{{{X_{h}({j\omega})}} - {{X_{heu}({j\omega})}}}{{X_{h}({j\omega})}} & {{{if}\mspace{14mu} {{X_{heu}({j\omega})}}} < {{X_{h}({j\omega})}}}\end{matrix}.} \right.}}} & (19)\end{matrix}$ where X_(h)(jω) is the integral admittance of a humanjoint and X_(heu)(jω) is the integral admittance of a coupledhuman-exoskeleton system.
 7. The resistive exoskeleton control system ofclaim 1, wherein the assistance ratio is defined by $\begin{matrix}{\mspace{79mu} {{ = {\frac{1}{\omega_{f}}{\int_{0}^{w_{f}}{{{\mathcal{F}}(\omega)}\ {{\omega}.\mspace{14mu} {wherein}}}}}}{{{\mathcal{F}}(\omega)} = \left\{ {\begin{matrix}\frac{{{X_{heu}({j\omega})}} - {{X_{h}({j\omega})}}}{{X_{h}({j\omega})}} & {{{if}\mspace{14mu} {{X_{heu}({j\omega})}}} \geq {{X_{h}({j\omega})}}} \\0 & {{{if}\mspace{14mu} {{X_{heu}({j\omega})}}} < {{X_{h}({j\omega})}}}\end{matrix},} \right.}}} & (20)\end{matrix}$ where X_(h)(jω) is the integral admittance of a humanjoint and X_(heu)(jω) is the integral admittance of a coupledhuman-exoskeleton system.
 8. A resistive exoskeleton control systemcomprising: a controller shaping a closed loop integral admittance of acoupled human exoskeleton system, wherein a frequency response magnitudeof the closed loop integral admittance is lower than that of a naturalhuman joint and generating an assistance ratio of approximately zero fordesired frequencies of interest, wherein the controller being stable andpassive.
 9. The resistive exoskeleton control system of claim 8, whereinthe controller generating a loop transfer function having a gain margingreater than
 1. 10. The resistive exoskeleton control system of claim 8,wherein a phase of the closed loop integral admittance is defined by/X_(heu)(jω)∈[−180°, 0°]∀ω.
 11. The resistive exoskeleton control systemof claim 8, wherein the controller generating a control transferfunction defined by${{U_{e}(s)} = \frac{{K_{\alpha}s^{2}} + {K_{\omega}s} + K_{\theta}}{s}},,$wherein K_(o)=I_(e)−I_(e) ^(d), K_(ω)=b_(e)−b_(e) ^(d), andK_(θ)=k_(e)−k_(e) ^(d) are the feedback gains on angular acceleration{umlaut over (θ)}_(e), angular velocity {dot over (θ)}_(e) and angleθ_(e) of an exoskeleton device.
 12. The resistive exoskeleton controlsystem of claim 8, wherein the controller generating a positiveresistance ratio over the desired frequencies of interest, wherein thepositive resistance defined by $\begin{matrix}{\mspace{79mu} {{ = {\frac{1}{\omega_{f}}{\int_{0}^{w_{f}}{{{\mathcal{F}}(\omega)}\ {{\omega}.\mspace{14mu} {wherein}}}}}}{{{\mathcal{F}}(\omega)} = \left\{ {\begin{matrix}0 & {{{if}\mspace{14mu} {{X_{heu}({j\omega})}}} \geq {{X_{h}({j\omega})}}} \\\frac{{{X_{h}({j\omega})}} - {{X_{heu}({j\omega})}}}{{X_{h}({j\omega})}} & {{{if}\mspace{14mu} {{X_{heu}({j\omega})}}} < {{X_{h}({j\omega})}}}\end{matrix}.} \right.}}} & (19)\end{matrix}$ where X_(h)(jω) is the integral admittance of a humanjoint and X_(heu)(jω) is the integral admittance of a coupledhuman-exoskeleton system.
 13. The resistive exoskeleton control systemof claim 8, wherein the assistance ratio is defined by $\begin{matrix}{\mspace{79mu} {{ = {\frac{1}{\omega_{f}}{\int_{0}^{w_{f}}{{{\mathcal{F}}(\omega)}\ {{\omega}.\mspace{14mu} {wherein}}}}}}{{{\mathcal{F}}(\omega)} = \left\{ {\begin{matrix}\frac{{{X_{heu}({j\omega})}} - {{X_{h}({j\omega})}}}{{X_{h}({j\omega})}} & {{{if}\mspace{14mu} {{X_{heu}({j\omega})}}} \geq {{X_{h}({j\omega})}}} \\0 & {{{if}\mspace{14mu} {{X_{heu}({j\omega})}}} < {{X_{h}({j\omega})}}}\end{matrix},} \right.}}} & (20)\end{matrix}$ where X_(h)(jω) is the integral admittance of a humanjoint and X_(heu)(jω) is the integral admittance of a coupledhuman-exoskeleton system.
 14. An exoskeleton control system comprising:a controller generating a positive resistance and approximately zeroassistance by shaping a closed loop integral admittance of a coupledhuman exoskeleton system over a desired frequency range, wherein thecontroller being stable and passive.
 15. The exoskeleton control systemof claim 14, wherein a frequency response magnitude of the closed loopintegral admittance is lower than that of a natural human joint over thedesired frequency range.
 16. The resistive exoskeleton control system ofclaim 14, wherein the controller generating a loop transfer function hasa gain margin greater than
 1. 17. The resistive exoskeleton controlsystem of claim 14, wherein a phase of the closed loop integraladmittance is defined by /X_(heu)(jω)∈[−180°, 0°]∀ω.
 18. The resistiveexoskeleton control system of claim 14, wherein the controllergenerating a control transfer function defined by${{U_{e}(s)} = \frac{{K_{\alpha}s^{2}} + {K_{\omega}s} + K_{\theta}}{s}},,$wherein K_(o)=I_(e)−I_(e) ^(d), K_(ω)=b_(e)−b_(e) ^(d), andK_(θ)=k_(e)−k_(e) ^(d) are the feedback gains on angular acceleration{umlaut over (θ)}_(e), angular velocity {dot over (θ)}_(e) and angleθ_(e) of an exoskeleton device.
 19. The resistive exoskeleton controlsystem of claim 14, wherein the controller generating a positiveresistance ratio over the desired frequencies of interest, wherein thepositive resistance ratio defined by $\begin{matrix}{\mspace{79mu} {{ = {\frac{1}{\omega_{f}}{\int_{0}^{w_{f}}{{{\mathcal{F}}(\omega)}\ {{\omega}.\mspace{14mu} {wherein}}}}}}{{{\mathcal{F}}(\omega)} = \left\{ {\begin{matrix}0 & {{{if}\mspace{14mu} {{X_{heu}({j\omega})}}} \geq {{X_{h}({j\omega})}}} \\\frac{{{X_{h}({j\omega})}} - {{X_{heu}({j\omega})}}}{{X_{h}({j\omega})}} & {{{if}\mspace{14mu} {{X_{heu}({j\omega})}}} < {{X_{h}({j\omega})}}}\end{matrix}.} \right.}}} & (19)\end{matrix}$ where X_(h)(jω) is the integral admittance of a humanjoint and X_(heu)(jω) is the integral admittance of a coupledhuman-exoskeleton system.
 20. The resistive exoskeleton control systemof claim 14, wherein the assistance ratio is defined by $\begin{matrix}{\mspace{79mu} {{ = {\frac{1}{\omega_{f}}{\int_{0}^{w_{f}}{{{\mathcal{F}}(\omega)}\ {{\omega}.\mspace{14mu} {wherein}}}}}}{{{\mathcal{F}}(\omega)} = \left\{ {\begin{matrix}\frac{{{X_{heu}({j\omega})}} - {{X_{h}({j\omega})}}}{{X_{h}({j\omega})}} & {{{if}\mspace{14mu} {{X_{heu}({j\omega})}}} \geq {{X_{h}({j\omega})}}} \\0 & {{{if}\mspace{14mu} {{X_{heu}({j\omega})}}} < {{X_{h}({j\omega})}}}\end{matrix},} \right.}}} & (20)\end{matrix}$ where X_(h)(jω) is the integral admittance of a humanjoint and X_(heu)(jω) is the integral admittance of a coupledhuman-exoskeleton system.